212 XII! INTEGRATION Assume that inf f(x) ^(U */)0>) f£ sup f(x) for all y e Y and all/e ^R(X). Show that under these conditions there exists a vaguely continuous mapping a: y\-~*ay of Y into M+(X) such that ay is carried by Tr~"l(.y) for all y e Y, and such that (U -/)(7r(.x)) = </, aw(X)> for all x e X, Consider the converse. (Consider the transpose '£/: MR(Y)^MR(X) of U (12.15.3).) 9. Let X be a compact space, let p. be a diffuse positive measure on X such that p,(X) = 1, and let u: X->X be a bijection such that both u and w"1 are /x-measurable, p, is u- invariant (Section 13.9, Problem 24) and u is ergodic with respect to (JL (Section 13.9, Problems 13 and 24). Show that for each e > 0 and each integer n > 0, there exists a ^-measurable subset A of X such that (i) the sets uJ(A) (0 gy <| n — 1) are pairwise disjoint, and (ii) the complement of (J uj(A) has measure g £ (Rokhlin's theorem). (Choose a ^-measurable set B such that 0 < ju,(B) < e//i, and construct the correspon- ding "Kakutani skyscraper" consisting of the sets up(Em) for m ^ 1 and 0 ^p < m (Section 13.9, Problem 14(d)). Show that we may take A to be the union of the sets ) for all integers j > 0 and integers m such that m ^ (/ + ; 19. SUPPORT OF A MEASURE. MEASURES WITH COMPACT SUPPORT (13.19.1) If n is any measure on X, the union of all ju-negligible open sets is \L-negligible (and hence is the largest ^-negligible open set). For if U is this union, it follows from (1 3.1 .9) that the measure induced by z on U is zero. The complement in X of the largest /^-negligible open set is called the support of n, and is denoted by Supp(^). To say that x e Supp(//) signifies that lA*l*00>0 for every neighborhood V of x, or equivalently (13.5.1) that M(l/l) > 0 for every /e Jfc(X) such that/(jc) ^ 0; or equivalently again, that for each neighborhood V of x there exists a function /e Jf C(X) with support contained in V, such that ju(/) ^ 0. If Supp(/x) = X, the only ^negli- gible continuous function is the constant 0. From the definition we have Supp(#) = Supp(|ju|), and it is clear that Supp(^) = Supp(/i) for all scalars a ^ 0. More generally, if g is any locally |ji|-integrable function, then we have Supp(# • //) c Supp(^) n Supp^u) ; for if we put v = g • |/4 and if an open set U does not intersect Supp(#) or does not intersect SuppOO, then |v|*(U) = 0 (13.14.1). (13.19.2) (i) If [i and v are positive measures, then Supp(/j + v) = SuppOu) u Supp(v).ists a continuous mapping TT of X onto I = [0,1]